# The mean of the commute time to work for a resident of a certain city is 28.3 minutes. Assume that the standard deviation of the commute time is 7.6 minutes to complete parts​ (a) through​ (c).a). what minimum percentage of commuter in the city has a commute time within 2 standard deviations of the mean?

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The mean of the commute time to work for a resident of a certain city is 28.3 minutes. Assume that the standard deviation of the commute time is 7.6 minutes to complete parts​ (a) through​ (c).

a). what minimum percentage of commuter in the city has a commute time within 2 standard deviations of the mean?

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Step 1

A normal distribution can be visualized by looking at the curve below.

X-axis shows Z score which is the distance from mean in terms of standard deviation. A z score of 2 would mean that the distance of the data point from the mean is equal to 2 standard deviation.

Commute time within 2 standard deviations of the mean would mean all communte times between Z score of -2 and +2 as shown by the shaded portion under the curve.

Step 2

The area under the curve represents probability or percentage or proportion. Since the value of probability ranges from 0 to1, the total area under the curve is always equal to 1 or 100%.

To calculate area percentage or probability corresponding to any Z score , Z tables can be used which yields percentage of area to the left of a score

We are interested in P(-2<Z<2) which will be equal to P(Z<2) - P(Z<-2)

This can be represented graphically as shown below.

Step 3

Let's see how Z table can be read for a Z score of -2 and +2

To find P(Z<=-2) , we have to find the value that lies at the intersection of row -2 and column .00

(NOTE: If this was P(Z<=-1.57), then we would look for value at the intersec...

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